Friday, May 24, 2013

I see a clock, but I cannot envision the clockmaker. The human mind is
unable to conceive of the four dimensions, so how can it conceive of a
God, before whom a thousand years and a thousand dimensions are as one?

The phrase of course stuck in my mind. Who is the clockmaker. I was more at ease with what Einstein quote spoke about with regards to the fourth dimension and here, thoughts of Dali made their way into my head.

In the United States, starting in the 1960s, creationists
revived versions of the argument to dispute the concepts of evolution
and natural selection, and there was renewed interest in the watchmaker
argument.

I have always shied away from the argument based on the analogy, fallacy and argument, as I wanted to show my thoughts here regardless of what had been transmitted and exposed on an objective level argument. Can I do this without incurring the wrought of a perspective in society and share my own?

I mean even Dali covered the Tesseract by placing Jesus on the cross in a sense Dali was exposing something that such dimensional significance may have been implied as some degree of Einstein's quote above? Of course I speculate but it always being held to some idea of a dimensional constraint that no other words can speak of it other then it's science. Which brings me back to Einstein's quote.

The construction of a hypercube can be imagined the following way:

1-dimensional: Two points A and B can be connected to a line, giving a new line segment AB.

2-dimensional: Two parallel line segments AB and CD can be connected to become a square, with the corners marked as ABCD.

3-dimensional: Two parallel squares ABCD and EFGH can be connected to become a cube, with the corners marked as ABCDEFGH.

4-dimensional: Two parallel cubes ABCDEFGH and IJKLMNOP can be connected to become a hypercube, with the corners marked as ABCDEFGHIJKLMNOP.

So for me it is about what lays at the basis of reality as to question that all our experiences, in some way masks the inevitable design at a deeper level of perceptions so as to say that such a diagram is revealing.

Also too if I were to deal with the subjectivity of our perceptions then how could I ever be clear as I muddy the waters of such straight lines and such with all the pictures of a dream by Pauli? I ask that however you look at the plainness of the dream expanded by Jung, that one consider the pattern underneath it all. I provide 2 links below for examination.

The Schläfli symbol notation describes every regular polytope, and is used widely below as a compact reference name for each.

The regular polytopes are grouped by dimension and subgrouped by
convex, nonconvex and infinite forms. Nonconvex forms use the same
vertices as the convex forms, but have intersecting facets. Infinite forms tessellate a one lower dimensional Euclidean space.

Infinite forms can be extended to tessellate a hyperbolic space.
Hyperbolic space is like normal space at a small scale, but parallel
lines diverge at a distance. This allows vertex figures to have negative
angle defects, like making a vertex with 7 equilateral triangles and allowing it to lie flat. It cannot be done in a regular plane, but can be at the right scale of a hyperbolic plane.

For most people I am sure it is of little interest that such an abstract language could have ever amounted to anything,since we might have been circumscribed to the natural living that is required that we could do without it. But really, can we?

So of course one appreciates those who start the conversation to help raise the questions in ones own mind. Might it be a shared response to something existing deeper in our society that it would warrant descriptions that we might be lacking in. Ways in which to describe something about nature. There is something definitely to be said about the geometer that can visualize the spaces within which they are working. It has to make sense. It has to describe something? Why then not just plain English(whatever language you choose)

String theory's mathematical tools were designed to unlock the most profound secrets of the cosmos, but they could have a far less esoteric purpose: to tease out the properties of some of the most complex yet useful types of material here on Earth.What Good are Mathematics in the Real World?

The language of physics
is mathematics. In order to study physics seriously, one needs to learn
mathematics that took generations of brilliant people centuries to work
out. Algebra, for example, was cutting-edge mathematics when it was being
developed in Baghdad in the 9th century. But today it's just the first
step along the journey.Guide to math needed to study physics

How mathematics arose from cognitive realizations. Ex. Newton and Calculus. The branches of mathematics. Who are it's developers and what did they develop and why?

It may be as important as the history in relation to how one may perceive the history and development of mathematics. These were important insights into the way one might of asked how did emergence exist if such things could have been imagined in the mind of the beholder. To attempt to describe nature in the way that one might do by invention? So are these mathematical things discovered or are they invented? Why the history is important?

This is the basis of the question of what already exists in terms of information has always existed and we are only getting a preview of a much more complicated system. It does not have to be a question of what a MBT exemplifies in itself, but raises the questions about what already exists, exists as part of what always existed. Where do ideas and mathematics come from?

This is a foundation stance that is taken right throughout science?
If it exists in the universe, it exists in you? How does one connect?

Thursday, April 19, 2012

"...underwriting the form
languages of ever more domains of mathematics is a set of deep patterns
which not only offer access to a kind of ideality that Plato claimed to
see the universe as created with in the Timaeus; more than this, the
realm of Platonic forms is itself subsumed in this new set of design
elements-- and their most general instances are not the regular solids,
but crystallographic reflection groups. You know, those things the
non-professionals call . . . kaleidoscopes! * (In the next exciting
episode, we'll see how Derrida claims mathematics is the key to freeing
us from 'logocentrism'-- then ask him why, then, he jettisoned the
deepest structures of mathematical patterning just to make his name...)

* H. S. M. Coxeter, Regular Polytopes (New
York: Dover, 1973) is the great classic text by a great creative force
in this beautiful area of geometry (A polytope is an n-dimensional
analog of a polygon or polyhedron. Chapter V of this book is entitled
'The Kaleidoscope'....)"

I just wanted to show you what has been physically reproduced in cultures. This in order to highlight some of the things that were part of our own make up, so you get that what has transpired in our societies has been part of something hidden within our own selves.

As I have said before it has become something of an effort for me to cataloged knowledge on some of the things I learn. The ways in which to keep the information together. I am not saying everyone will do this in there own way but it seems to me that as if some judgement about our selves is hidden in the way we had gathered information about our own lives then it may have been put together like some kaleidoscope.

So to say then past accomplishments were part of the designs, what had we gained about our own lives then? What page in the book of Mandalas can you have said that any one belonged to you? It was that way for me in that I saw the choices. These I thought I had built on my own, as some inclination of a method and way to deliver meaning into my own life. Then through exploration it seem to contain the energy of all that I had been before as to say that in this life now, that energy could unfold?

Scan of painting 19th century Tibetan Buddhist thangka painting

Maṇḍala (मण्डल) is a Sanskrit word meaning "circle." In the Buddhist and Hindu religious traditions their sacred art
often takes a mandala form. The basic form of most Hindu and Buddhist
mandalas is a square with four gates containing a circle with a center point. Each gate is in the shape of a T.[1][2] Mandalas often exhibit radial balance.[3]

In various spiritual traditions, mandalas may be employed for
focusing attention of aspirants and adepts, as a spiritual teaching
tool, for establishing a sacred space, and as an aid to meditation and trance induction. According to the psychologist David Fontana,
its symbolic nature can help one "to access progressively deeper levels
of the unconscious, ultimately assisting the meditator to experience a
mystical sense of oneness with the ultimate unity from which the cosmos
in all its manifold forms arises."[6] The psychoanalystCarl Jung saw the mandala as "a representation of the unconscious self,"[citation needed] and believed his paintings of mandalas enabled him to identify emotional disorders and work towards wholeness in personality.[7]

In common use, mandala has become a generic term for any plan, chart or geometric pattern that represents the cosmos metaphysically or symbolically, a microcosm of the Universe from the human perspective.[citation needed]

So what does this mean then that you see indeed some subjects that are allocated toward design of to say that it may be an art of a larger universal understanding that hidden in our natures the will to provide for something schematically inherent? Our nature, as to the way in which we see the world. The way in which we see science. What cosmic plan then to say the universe would unfold this way, or to seek the inner structure and explanations as to the way the universe began. The way we emerged into consciousness of who you are?

The kaleidoscope was perfected by Sir
David Brewster, a Scottish scientist, in 1816. This technological
invention, whose function is literally the production of beauty, or
rather its observation, was etymologically a typical aesthetic form of
the nineteenth century - one bound up with disinterested contemplation. (The etymology of the word is formed from kalos (beautiful), eidos (form) and scopos (watcher) - "watcher of beautiful shapes".)
The invention is enjoying a second life today - as the model for many
contemporary abstract works. In Olafur Eliasson's Kaleidoscope (2001),
the viewer takes the place of the pieces of glass, producing a myriad of
images. In an inversion of the situation involved in the classic
kaleidoscope, the watcher becomes the watched. In Jim Drain's
Kaleidoscope (2003), the viewer is also plunged physically inside the
myriad of abstract forms, and his image becomes a part of the
environment. Spin My Wheel (2003), by Lori Hersberger, also forms a
painting that is developed in space, spilling beyond the frame of the
picture, its projected image constantly changing, dissolving the
surrounding world with an infinite play of reflections in fragments of
broken mirror. The viewer becomes one of the subjects of the piece. (Not
the subject, as in Eliasson's work, but one of its subjects.)See: The End of Perspective-Vincent Pécoil.

Would there be then some algorithmic style to the code written in your life as to have all the things you are as some pattern as to the way in which you will live your life? I ask then what would seem so strange that you might not paint a picture of it? Not encode your life in some mathematical principle as to say that life emerge for you in this way?

Although Aristotle in
general had a more empirical and experimental attitude than Plato,
modern science did not come into its own until Plato's Pythagorean confidence in the mathematical nature of the world returned
with Kepler, Galileo, and Newton. For instance, Aristotle, relying on a
theory of opposites that is now only of historical interest, rejected
Plato's attempt to match the Platonic Solids with the elements -- while
Plato's expectations are realized in mineralogy and crystallography,
where the Platonic Solids occur naturally.Plato and Aristotle, Up and Down-Kelley L. Ross, Ph.D.

Friday, June 10, 2011

"I’m a Platonist — a follower of Plato — who believes that one didn’t invent these sorts of things, that one discovers them. In a sense, all these mathematical facts are right there waiting to be discovered."Harold Scott Macdonald (H. S. M.) Coxeter

While a layman in my pursuance and understanding of the nature of geometry, it is along the way we meet some educators who fire up our excitement. For me it is about the truth of what lies so close to the soul's ideal.

To me this is one of the greatest achievements of mathematical structures that one could encounter, It revolutionize many a view, that been held to classical discriptions of reality.

In the quiet achievement of Riemann’s tutorial teacher Gauss, recognized the great potential in his student. On the curvature parameters, we recognize in Gauss’s work, what would soon became apparent? That we were being lead into another world for consideration?

XXV. Gaussian Co-ordinates-click on Picture

Albert Einstein (1879–1955). Relativity: The Special and General Theory. 1920.

So here we are, that we might in our considerations go beyond the global perspectives, to another world that Einstein would so methodically reveal in the geometry and physics, that it would include the electromagnetic considerations of Maxwell into a cohesive whole and beyond.

Tuesday, January 20, 2009

After some thinking, how is it one can think that such an abstraction could descend into the modern mind, and think it reveals the idea of nature in expression? It's just an artistic interpretation then, nothing more?

The cycle of creativity is an interesting circuit to the droplet form. As an idea and such, a condensible feature of the "subtle and storm gathering" into a distillation of a kind?

: If you keep pulling the hypercube into higher and higher dimensions you get a polytope. Coxeter is famous for his work on regular polytopes. When they involve coordinates made of complex numbers they are called complex polytopes.

How many crack point numbers for this poetic version?:) Baez's Crackpot index is a joke.

Sunday, June 08, 2008

At this point in the development, although geometry provided a common framework for all the forces, there was still no way to complete the unification by combining quantum theory and general relativity. Since quantum theory deals with the very small and general relativity with the very large, many physicists feel that, for all practical purposes, there is no need to attempt such an ultimate unification. Others however disagree, arguing that physicists should never give up on this ultimate search, and for these the hunt for this final unification is the ‘holy grail’. Michael Atiyah

"No Royal Road to Geometry?"

Click on the Picture

Are you an observant person? Look at the above picture. Why ask such a question as to, "No Royal Road to Geometry?" This presupposes that a logic is formulated that leads not only one by the "phenomenological values" but by the very principal of logic itself.

All those who have written histories bring to this point their account of the development of this science. Not long after these men came Euclid, who brought together the Elements, systematizing many of the theorems of Eudoxus, perfecting many of those of Theatetus, and putting in irrefutable demonstrable form propositions that had been rather loosely established by his predecessors. He lived in the time of Ptolemy the First, for Archimedes, who lived after the time of the first Ptolemy, mentions Euclid. It is also reported that Ptolemy once asked Euclid if there was not a shorter road to geometry that through the Elements, and Euclid replied that there was no royal road to geometry. He was therefore later than Plato's group but earlier than Eratosthenes and Archimedes, for these two men were contemporaries, as Eratosthenes somewhere says. Euclid belonged to the persuasion of Plato and was at home in this philosophy; and this is why he thought the goal of the Elements as a whole to be the construction of the so-called Platonic figures. (Proclus, ed. Friedlein, p. 68, tr. Morrow)

I don't think I could of made it any easier for one, but to reveal the answer in the quote. Now you must remember how the logic is introduced here, and what came before Euclid. The postulates are self evident in his analysis but, little did he know that there would be a "Royal Road indeed" to geometry that was much more complex and beautiful then the dry implication logic would reveal of itself.

It's done for a reason and all the geometries had to be leading in this progressive view to demonstrate that a "projective geometry" is the final destination, although, still evolving?

If you are "ever the artist" it is good to know in which direction you will use the sun, in order to demonstrate the shadowing that will go on into your picture. While you might of thought there was everything to know about Plato's cave and it's implication I am telling you indeed that the logic is a formative apparatus concealed in the geometries that are used to explain such questions about, "the shape of space."

Polytopes and allotrope are examples to me of "shapes in their formative compulsions" that while very very small in their continuing expression, "below planck length" in our analysis of the world, has an "formative structure" in the case of the allotrope in the material world. The polytopes, as an abstract structure of math thinking about the world. As if in nature's other ways.

Sean Carroll:But if you peer closely, you will see that the bottom one is the lopsided one — the overall contrast (representing temperature fluctuations) is a bit higher on the left than on the right, while in the untilted image at the top they are (statistically) equal. (The lower image exaggerates the claimed effect in the real universe by a factor of two, just to make it easier to see by eye.)

“I’m a Platonist — a follower of Plato — who believes that one didn’t invent these sorts of things, that one discovers them. In a sense, all these mathematical facts are right there waiting to be discovered.”Harold Scott Macdonald (H. S. M.) Coxeter

Moving to polytopes or allotrope seem to have values in science? Buckminister Fuller and Richard Smalley in terms of allotrope.

I was looking at Sylvestor surfaces and the Clebsch diagram. Cayley too. These configurations to me were about “surfaces,” and if we were to allot a progression to the “projective geometries” here in relation to higher dimensional thinking, “as the polytope[E8]“(where Coxeter[I meant to apologize for misspelling earlier] drew us to abstraction to the see “higher dimensional relations” toward Plato’s light.)

As the furthest extent of the Conjecture , how shall we place the dynamics of Sylvestor surfaces and B Fields in relation to the timeline of these geometries? Historically this would seem in order, but under the advancement of thinking in theoretics does it serve a purpose? Going beyond “planck length” what is a person to do?

Thanks for the clarifications on Lagrange points. This is how I see the WMAP.

Diagram of the Lagrange Point gravitational forces associated with the Sun-Earth system. WMAP orbits around L2, which is about 1.5 million km from the Earth. Lagrange Points are positions in space where the gravitational forces of a two body system like the Sun and the Earth produce enhanced regions of attraction and repulsion. The forces at L2 tend to keep WMAP aligned on the Sun-Earth axis, but requires course correction to keep the spacecraft from moving toward or away from the Earth.

Such concentration in the view of Sean’s group of the total WMAP while finding such a concentration would be revealing would it not of this geometrical instance in relation to gravitational gathering or views of the bulk tendency? Another example to show this fascinating elevation to non-euclidean, gravitational lensing, could be seen in this same light.

Such mapping would be important to the context of “seeing in the whole universe.”

Tuesday, February 12, 2008

Although Aristotle in general had a more empirical and experimental attitude than Plato, modern science did not come into its own until Plato's Pythagorean confidence in the mathematical nature of the world returned with Kepler, Galileo, and Newton. For instance, Aristotle, relying on a theory of opposites that is now only of historical interest, rejected Plato's attempt to match the Platonic Solids with the elements -- while Plato's expectations are realized in mineralogy and crystallography, where the Platonic Solids occur naturally.Plato and Aristotle, Up and Down-Kelley L. Ross, Ph.D.

This is the first introduction then that is very important to me about what is perceived as a mathematical framework. So it is not such an effort to think about our world and think hmmmm.... a mathematical abstract of our reality is there to be discovered. I first noticed this attribute in Pascal's triangle.

The sudden shrinking of Euclidean geometry to a subspecies of the vast family of mathematical theories of space shattered some illusions and prompted important changes in our the philosophical conception of human knowledge. Thus, for instance, after these nineteenth-century developments, philosophers who dream of a completely certain knowledge of right and wrong secured by logical inference from self-evident principles can no longer propose Euclidean geometry as an instance in which a similar goal has proved attainable. The present article reviews the aspects of nineteenth century geometry that are of major interest for philosophy and hints in passing, at their philosophical significance.

While I looked further into the world of Pythagorean developments I wondered how such an abstract could have ever lead to the world of non-euclidean geometries. There is this progression of the geometries that needed to be understood. It included so many people that we only now acknowledge the greatest names but it is in the exploration of "theoretical excellence" that we gain access to the spirituality's of the mathematical world.

"I’m a Platonist — a follower of Plato — who believes that one didn’t invent these sorts of things, that one discovers them. In a sense, all these mathematical facts are right there waiting to be discovered."Donald (H. S. M.) Coxeter

While some would wonder what value this exploration into such mathematical abstracts, how could we describe for ourselves the ways things would appear at such levels microscopically reduced, has an elemental quality to it? Yes, I have gone to one extreme, and understand, it included so many different mathematics, how could we ever understand this effort and assign it's rightful place in history? Theoretics then, is this effort?

How Strange the elements of our world?

The crystalline state is the simplest known example of a quantum , a stable state of matter whose generic low-energy properties are determined by a higher organizing principle and nothing else.Robert Laughlin

Saturday, September 22, 2007

"I’m a Platonist — a follower of Plato — who believes that one didn’t invent these sorts of things, that one discovers them. In a sense, all these mathematical facts are right there waiting to be discovered."Donald (H. S. M.) Coxeter

If I had thought there was a way to describe the "interior" of the blackhole, it would be by recognizing the dimensionality the blackhole had to offer. One had to know where to locate "this place in the natural world." If we had understood the energy values of the particle world colliding(that space and frame of reference, then what were we finding that such a place in dimensionality could exist in the natural world? Yoyu had to accept that there was dynamical moves that werre being defined as a possiility.

So what ways would allow us to do this, and this is part of the idea that came to me as I was thinking about the place where all possibilities could exist. Yet, what existed as "moduli form in the valleys" was being extended. So I am connecting other things here too.

Monday, March 19, 2007

"I’m a Platonist — a follower of Plato — who believes that one didn’t invent these sorts of things, that one discovers them. In a sense, all these mathematical facts are right there waiting to be discovered."Donald (H. S. M.) Coxeter

Clifford of Asymptotia drew our attention to this for examination and gives further information and links with which to follow.

He goes on to write,"Let’s not get carried away though. Having more data does not mean that you worked harder to get it. Mapping the human genome project involves a much harder task, but the analogy is still a good one, if not taken too far."

Of course since the particular comment of mine was deleted there, and of course I am okay with that. It did not mean I could not carry on here. It did not mean that I was not speaking directly to the way these values in dimensional perspective were not being considered.

There had to be a route to follow that would lead one to think in such abstract spaces. Of course, one does not want to be divorced from reality. So one should not think that because the geometry of GR is understood, that you think nothing can come from the microseconds after the universe came into expression.

At this point in the development, although geometry provided a common framework for all the forces, there was still no way to complete the unification by combining quantum theory and general relativity. Since quantum theory deals with the very small and general relativity with the very large, many physicists feel that, for all practical purposes, there is no need to attempt such an ultimate unification. Others however disagree, arguing that physicists should never give up on this ultimate search, and for these the hunt for this final unification is the ‘holy grail’. Michael Atiyah

The Holy Grail sure comes up lots doesn't it:) Without invoking the pseudoscience that Peter Woit spoke of. I thought, if they could use Babar, and Alice then I could use the Holy Grail?

Like Peter I will have to address the "gut feelings" and the way Clifford expressed it. I do not want to practise pseudoscience as Peter is about the landscape.:)

When ones sees the constituent properties of that Gossett polytope 421 in all it's colours, the complexity of that situation is quite revealing. Might we not think in the time of supergravity, gravity will become weak, in the matter constitutions that form.

As in Neutrino mixing I am asking you to think of the particles as sound as well as think them in relation to the Colour of Gravity. If you were just to see grvaity in it's colourful design and what value that gravity in face of the photon moving within this gravitational field?

The "geometry of curvature" had to be implied in the outcome, from that quantum world? Yet at it's centre, what is realized? You had to be lead there in terms of particle research to know that you are arriving at the "crossover point." The superfluid does this for examination.

So what chance do we have, if we did not think this geometry was attached to processes that would unfold into the bucky ball or the fullerene of science. To say that the outcome had a point of view that is not popular. I do not count myself as attached to any intelligent design agenda, so I hope people will think I do not care about that.

NATHAN MYHRVOLD

I found the email debate between Smolin and Susskind to be quite interesting. Unfortunately, it mixes several issues. The Anthropic Principle (AP) gets mixed up with their other agendas. Smolin advocates his CNS, and less explicitly loop quantum gravity. Susskind is an advocate of eternal inflation and string theory. These biases are completely natural, but in the process the purported question of the value of the AP gets somewhat lost in the shuffle. I would have liked more discussion of the AP directly

So all the while you see the complexity of that circle and how long it took a computer to map it, it has gravity in it's design, whether we like to think about it or not?

But of course we are talking about the symmetry and any thing less then this would have been assign a matter state, as if symmetrical breaking would have said, this is the direction you are going is what we have of earth?

Isostatic Adjustment is Why Planets are Round?

While one thinks of "rotational values" then indeed one would have to say not any planets is formed in the way the sun does. Yet, in the "time variable understanding" of the earth, we understand why it's shape is not exactly round.

Do you think the earth and moon look round if your were considering Grace?

On the moon what gives us perspective when a crater is formed to see it's geological structure? It's just not a concern of the mining industry, as to what is mined on other orbs, but what the time variable reveals of the orbs structure as well.

Monday, September 11, 2006

"I’m a Platonist — a follower of Plato — who believes that one didn’t invent these sorts of things, that one discovers them. In a sense, all these mathematical facts are right there waiting to be discovered."Harold Scott Macdonald (H. S. M.) Coxeter

Some would stop those from continuing on, and sharing the world behind the advancements in geometry. I am very glad that I can move from the Salvador Dali image of the crucifixtion, to know, that minds engaged in the "pursuites of ideas" as they may "descend from heaven," may see in a man like Donald Coxeter, the way and means to have ideas enter his mind and explode in sociological functions? Hmmmm. what does that mean?

Without stealing the limelight from Donald, I wanted to put the thinking of Michael Atiyah along side of him too. So you understand that those who speak about the "physics" have things underlying this process which help hold them to the very fabric of thinking.

Some do not know of "this geometric process" I speak, where such manifestation arise from the very essence of the thinking soul. If you began to learn about yourself you would know that such abstractions are much closer to the "pure thought" then any would have realized.

Some meditate to get to this essence. Some know, that in having gone through a journey of discovery that they will find the very patterns sealed within each of the souls.

How does it arise? You had to follow this journey through the "muddle maze" of the dreaming mind to know that patterns in you can direct the vision of things according to what you yourself already do inherently.

Now some of you "know," don't you, with regards to what I am saying? I spoke often of "Liminocetric structures" just to help you along, and help you realize that the sociological standing of exchange houses many forms of thinking that we had gained previously. Why as a soul of the "thinking mind" should you loose this part of yourself?

So you begin with the "Platonic Forms" and look for the soccer ball/football? THis process resides at many levels and Dirac was very instrumental in speaking about the basis of the geometer and his vision of things. Along side of course the algebraic way.

This is very real, and not so abstract that you may have departed form the real world to say, you have lost touch? Do you think only "in a square box" and cannot percieve anything beyond the "condensive thoughts and model apprehensions" which hold you to your own design?

Maybe? :)

But the world is vast in terms of discovery, that the question of mathematics again draws us back too, was "Mathematics invented or discovered?" So "this premise" as a question formed and with it "the roads" that lead to inquiry?

Al these forms of geometrics leading to question about "Quantum geometry" and how would such a cosmological world reveal to the thinkingmind "the microscopic" as part of the dynamical world of our everyday living?

Only a cynic casts the diversions and illusions to what is real. Because they cannot inherently deal with the "strange language of geometrics" that issues forth in model apprehensions. This is the basis from which Einstein solved the problems of his day.

But the question is what geometrics could ever reside at such a microscopic level?